Prompted by recent success in the research and application of earthquake early warning systems, we conducted an investigation into the relationship between the magnitudes of earthquakes and the properties of the first three seconds of the P waves at a single station. Using waveforms at station NACB of the Broadband Array in Taiwan for Seismology from 46 earthquakes within 100-km epicentral distance, we found a linear correlation between the magnitudes and the logarithmic characteristic periods of the initial P waves for earthquakes up to magnitude 6.5. We also detected a deterministic relationship between the magnitudes and the peak amplitudes of the initial P waves for earthquakes of magnitude M < 6.5. While further study needs to be done on the behavior of the initial P-wave peak amplitudes from larger earthquakes, we propose here that the characteristic periods and peak amplitudes of the first three seconds of the P waves be used jointly in onsite (single-station) earthquake early warning operations.